Affine transform of logistic map to a dynamic system

Question

I am reading the following technic report:

https://homepages.laas.fr/henrion/papers/odds.pdf

For the section 5 Example,

It gives a logistic map: $\bar{x}_{k+1} = 4\bar{x}_k(1-\bar{x}_k)$

and then define the affine change of variables $\bar{x}_k = \frac{1}{2}(1-\bar{x}_k)$

So it obtain discrete dynamic system $$f(x) = 2x^2-1$$

My questions are the following:

1. After plug in, I get $(1-x_k)(1+x_k) = 1-x^2_k$. I am not sure how to get that.
2. I am confused what is the difference of a "map" and "dynamic system". I think the logistic map is a dynamic system. Why does the author do the transformation?

thanks!

Answer 1

1. If you substitute $x_k=\frac{1}{2}(1-\bar{x}_k)$ (and $x_{k+1}=\frac{1}{2}(1-\bar{x}_{k+1})$), you will get $x_{k+1}=2x_k^2-1$ as requested. Note the indices!

2. Logistic map is the name for a specific dynamic system.